Biquad notch filter

ABSTRACT

Though using an operational amplifier having not so large GB product, in order to obtain a constituting circuit is given to have enough to a deep notch characteristics, a biquad filter comprising a first and a second stages consisting an inverse amplifier and a third and a forth stages consisting an inverse integrator, replacing a feed back resistor to impedance element for compensating. The compensating impedance element is used a reactance, for example, an inductor or capacitor connected to the feed back resistor.

BACKGROUND OF THE INVENTION

1. The present invention relates to a biquad type notch filter a kind of active filter for improvement having a deep notch characteristic.

DESCRIPTION OF THE RELATED ART

2. A notch filter is alias a band elimination filter or a band stop filter, for attenuating or damping a part of band in a pass-band. There are many types of active filter, however biquad type filter uses a constantly to have a characteristic adjusting easily.

3. A normal type biquad circuit constitutes first stage an add-inverse low pass circuit, second stage an integrate-inverse circuit, third stage an inverse circuit, and a feed back circuit of from the third stage output to the first stage input. When output of a notch filter is acquired, it is well known to be obtained using another outer adder weighing three signals, the first stage output and the second stage output or the third stage output, and adding them.

4. When Q is changed by notch filter of this type, a gain of the first stage is changed in proportionate to Q and simultaneously it may change a weight of a signal adding from the first stage output to the outer adder. However, it has troublesome that operating level of a following stage circuit is high.

5. To improve such trouble, an another type of biquad circuit is constituted that first stage is an add-inverse circuit, second is an add-inverse circuit too, third stage is an integrate-inverse circuit and forth stage is an integrate-inverse circuit too, and feed back circuits which is from the third stage output to the first stage input and from the forth stage output to the second stage input, respectively. When Q changes in this type of biquad circuit, only weight of signal is added an adder of from the first stage output to the second stage input, simultaneously it is changed in proportionate to Q. By this way, not only the circuit is easier than the former circuit is that gains of two stages must be changed, but the latter circuit has no trouble about operating level is high in proportionate to Q. However, the biquad circuit is that the first stage output is notch filter output. When Q enlarges in the circuit, a real operational amplifier (hereinafter, abbreviated “OP amp.”) is different from ideal model of this kind and a gain-bandwidth product (GB product) is a finite. Therefore, in many case, a heavy or deep notch is not obtained. When GB product seeks to be enlarged, it is difficult to be given a wide band and a high gain OP amp.

6. Aim of the present invention provides, according to afore-mentioned, in a biquad notch filter having double feed back circuits, when an OP amp is used, not so large GB product, constituted circuit is obtained having an enough deep notch characteristics.

SUMMARY OF THE INVENTION

7. In view of the foregoing problems, it is an object of the present invention to provide a biquad notch filter comprising:

8. a first and a second stages composing an inverse amplifier and a third and a forth stages composing an inverse integrator,

9. a feed back circuit connected from an output of the inverse integrator of the third stage to the inverse amplifier input of the first stage and from an output of the inverse integrator of the forth stage to the inverse amplifier input of the second stage, respectively,

10. said inverse amplifier composed of an operational amplifier and a feed back resistor between input and output of the amplifier,

11. said inverse integrator composed of an operational amplifier and a feed back capacitor between input and output of the amplifier,

12. empowered output of the inverse amplifier of the first stage being filter output,

13. a reactance element connected to said feed back resistor of said second stage corresponding to a gain and cut off frequency of said inverse amplifier constituted as an impedance element, and obtained a deep notch characteristic.

14. The biquad notch filter is preferred that said impedance element constituted that an inductor is connected in series to a feed back resistor of an inverse amplifier of said second stage.

15. The biquad notch filter is preferred that said impedance element constituted that a capacitor is connected in parallel to a feed back resistor of an inverse amplifier of said second stage.

16. The biquad notch filter is preferred that said impedance element constituted that a capacitor is connected to a feed back resistor in a parallel and an inductor is connected to them in series.

BRIEF DESCRIPTION OF THE DRAWINGS

17. The objects and features of the present invention will become more apparent from the consideration of the following detailed description taken in conjunction with the accompanying drawings, in which:

18.FIG. 1 is a structural circuit diagram of a biquad type notch filter having double feed back circuits used as the present invention;

19.FIG. 2 is a structural circuit diagram of an inverse amplifier and an inverse integrator which is a structural component of the circuit of FIG. 1;

20.FIG. 3 is a structural circuit diagram of an example of the present invention in which resistance element is replaced by impedance circuit for compensator in order to improve a notch characteristic of the circuit in FIG. 1;

21.FIG. 4 is a structural circuit diagram of impedance circuit for compensator using three structural methods of the present invention;

22.FIG. 5 is a diagram showing relation of open-loop gain and cut-off frequency of OP amp. gained deep notch using a compensator; and

23.FIG. 6 is a characteristic diagram showing an example of frequency characteristic using a compensator and no compensator.

DESCRIPTION OF THE PREFERRED EMBODIMENT

24. A biquad notch filter according to the present invention is as aforesaid, first and second stage is an inverse-amplifier circuit and third and forth stage is an inverse integrator circuit, respectively. When the circuit is used by an OP amp. which has not enough large GB product, the inverse-amplifier of the second stage constituted by a impedance element adding a reactance element, such as inductor or capacitor or the same. Therefore, it is obtained a quite deep notch characteristic by using them.

25. A detailed description will now be given of an example of a biquad notch filter according to the present invention based on the above-mentioned embodiment. In a biquad notch filter shown by FIG. 1, 1 is a signal input terminal, 2 is a notch filter output terminal, first and second stage is a inverse-amplifier 3,4, and third and forth stage is a inverse-integrator 5,6, respectively. However, using an impedance element selected any one of three kinds of compensator circuit shown by FIG. 4 composed R₂ and another reactance element, when OP amp. having not so large GB product is used, the circuit is obtained having an extremely deep notch characteristic. In FIG. 1, R₀-R₃ are resistor, C₀ is a capacitor, OP₁-OP₄ are OP amp., e₁ is an input signal, e₂ is a notch filter output signal and 7,8 are feed back lines or loops.

26. As structural circuit diagram of biquad notch filter of the present invention is quite simple as before mentioned, its characteristic will be described below. In the circuit shown by FIG. 1, when OP amp. of each circuit is ideal amplifier, notch frequency ω₀ is defined as 1/(R₃C₀) and Q is defined R₁/R₂. This circuit is basically constituted in combination of inverse-amplifier (a) and inverse-integrator (b). However, when transfer function H₀(s) of OP amp. itself, as shown by formula 1 is low pass filter of first approximation (open loop gain is A and cut off frequency is ω_(c)), transfer function H_(A)(s) of inverse amplifier (a) is defined as formula 2 and transfer function H_(r)(s) of inverse integrator (b) is defined as formula 3. $\begin{matrix} {{H_{0}(s)} = {{A\frac{\omega_{c}}{s + \omega_{c}}} = \frac{GB}{s + \omega_{c}}}} & (1) \\ {{H_{A}(s)} = {\frac{e_{b}}{e_{a}} = \frac{A\quad \omega_{c}\frac{R_{b}}{R_{a} + R_{b}}}{s + \left\{ {1 + {A\frac{R_{a}}{R_{a} + R_{b}}}} \right\}}}} & (2) \\ {{H_{1}(s)} = {\frac{e_{d}}{e_{c}} = {- \frac{A\quad \omega_{0}\omega_{c}}{s^{2} + {\left\{ {\omega_{0} + {\left( {1 + A} \right)\omega_{c}}} \right\} s} + {\omega_{0}\omega_{c}}}}}} & (3) \\ {{{where}\quad \omega_{0}} = \frac{1}{R_{3}C_{0}}} & \quad \end{matrix}$

27. Using these transfer function, if transfer function H_(N) (s) between terminal 1 and 2 of circuit shown by FIG. 1 is solved, it is obtained as formula 4. $\begin{matrix} {{H_{N}(s)} = {\frac{e_{2}}{e_{1}} = {{- \frac{a}{b}} \cdot \frac{{sedg}^{2} + {cdml}^{2}}{{cedg}^{2} + {aeghl} + {cdml}^{2}}}}} & (4) \\ {where} & \quad \\ {{a = \frac{A\quad \omega_{c}}{2}},{b = {s + \frac{A\quad \omega_{c}}{2}}},{c = {s + {\left( {1 + {A\frac{R_{1}}{R_{1} + R_{2}}}} \right)\omega_{c}}}},{d = {s + {\left( {1 + \frac{A}{2}} \right)\omega_{0}}}},{l = {A\quad \omega_{0}\omega_{c}}},{m = {A\quad \omega_{c}\frac{R_{2}}{R_{0} + R_{2}}}},{e = {s + {\left( {1 + {A\frac{R_{0}}{R_{0} + R_{2}}}} \right)\omega_{c}}}},{g = {{s^{2} + {\left\{ {\omega_{0} + {\left( {1 + A} \right)\omega_{c}}} \right\} s} + {\omega_{0}\omega_{c}\quad {and}\quad h}} = {A\quad \omega_{0}\frac{R_{2}}{R_{1} + R_{2}}}}}} & \quad \end{matrix}$

28. In formula 4, as numerator is 7th order and denominator is 8 th order, it is too high order to analyze easily. However, as formula 2 and 3 are first order and second order, respectively, when an OP amp. amplification factor A and a cut off frequency ω_(c) are not so large, then it occurs a phase rotating by them. Therefore, it predicts that is not sufficiently noise canceling by inverse amplifier of the first stage shown by FIG. 1.

29. A position which is compensating the phase rotating is predicted that R₂ position is a feed back resistor of second stage shown by FIG. 1 is the optimum position. Since changing impedance of only one position is influenced two feed back loops 7,8, therefore the position is that is in all over the circuit influenced.

30. A circuit in which R₂ is replaced by compensator element of impedance element Z, is shown by FIG. 3. In FIG. 3, three kinds of Zs of inner constitution Z_(a) to Z_(c), are shown by FIG. 4(A) to FIG. 4(C). In Z_(a), an inductor L is connected to a feed back resistor R₂ in series, in Z_(b), a capacitor C₁ is connected to R₂ in parallel and in Z_(c), a capacitor C₁ is connected R₂ in a parallel to R₂ and an inductor L is connected to them in series.

31. By using these three kinds of compensator, it is confirmed by calculation of absolute value in formula 4. Obtained the calculation is shown by FIG. 5.

32. Longitudinal axis of FIG. 5 is an open loop gain A of OP amp. and transverse axis is a cut off frequency f_(c) (ω_(c) =2π f _(c)) of first approximation low-pass. In FIG. 5, a circle (◯) shows that compensating is performed using a Z_(a),  shows that compensating is performed using a Z_(b) and ⊚ shows that compensating is performed using a Z_(c), respectively. A described position on coordinate are A and f_(c) used as the calculation. Another parameter of calculation about formula 4 are ω₀=2×π×5×10³, R₀=10⁴, R₁=4×10⁵ and R₂=10⁴. As Q is R₁/R₄, so Q is 40.

33. As shown by FIG. 5, when A is relatively large and f_(c) is relatively low, it can compensate by circuit shown by FIG. 4(b). Accordingly, it is approved that by the circuit as shown by FIG. 4(c) obtained in combination of FIG. 4(a) and (b), in whole limits of OP amp. can compensate.

34. In examples of calculation, the above ◯ shown by FIG. 5 when compensating circuit is not used, input signal of 5 kHz is damped only 26 dB. Most deep notch point arises in 4.9735 kHz, though damping of the point is only 21.316 dB. However, when compensating circuit Z_(a) is used and L is selected 6.3009 . . . ×10⁻⁴ H(Henry), then mostly perfect zero point is produced having a damping factor 170 dB or more at frequency 4.9974 . . . kHz. In the above  shown by FIG. 5, when there is only keeping R₂, it is obtained that a damping factor is only 0.17 dB at the point of frequency 5 kHz, and no notch is produced. However, when compensating circuit Z_(b) is used, and C₁ is selected in 9.02465 . . . ×10⁻¹¹ F, then mostly perfect zero point is produced having a damping factor 100 dB or more at frequency 4.0610455 . . . kHz.

35. Furthermore, in the above c=⊚ shown by FIG. 5, when there is only keeping R₂, notch doesn't arise. However, using a compensating circuit Z_(c) and C₁ is selected in 1.20007 . . . ×10⁻¹¹ F and L is selected in 5.683497 . . . ×10⁻⁴ H, then notch is obtained having a dumping factor 110 dB or more at frequency of 4.70643162 . . . kHz. The compensating circuit Z_(c) can change both L and C₁, so combination of the L and C₁ e are infinity. Therefore, it can be corresponding to a combination of A and ω_(c) of OP amp. through in high extent limits.

36. Furthermore, in an example of using the compensating circuit, when a compensating circuit Z_(b) is used, obtained characteristic frequency of a dumping factor α is shown by FIG. 6. In FIG. 6, dotted line is that the case of no compensating circuit is used. When a compensating circuit is used, a deep notch characteristic is obtained as shown in a solid line.

37. As described above, the present invention is to provide a biquad notch filter having double feed back circuit, using an OP amp. not so large GB product, that constituting circuit is given to have enough to a deep notch characteristics. Therefore, a difficulty of selecting OP amp. is expunged.

38. While the present invention has been described with reference to the particular illustrative embodiments, it is not to be restricted by this embodiment but only by the appended claims. It is to be appreciated that those skilled in the art can change or modify the embodiments without departing from the scope and spirit of the present invention. 

What is claimed is:
 1. A biquad notch filter comprising: a first and a second stages composing an inverse amplifier and a third and a forth stages composing an inverse integrator, a feed back circuit connected from an output of the inverse integrator of the third stage to the inverse amplifier input of the first stage and from an output of the inverse integrator of the forth stage to the inverse amplifier input, respectively, said inverse amplifier composed of an operational amplifier and a feed back resistor between input and output of the amplifier, said inverse integrator composed of an operational amplifier and a feed back capacitor between input and output of the amplifier, empowered output of the inverse amplifier of the first stage to filter output, a reactance element connected to said feed back resistor of said second stage corresponding to a gain and cut off frequency of said inverse amplifier constituted as an impedance element, and obtained a deep notch characteristic.
 2. The biquad notch filter claimed in claim 1 , wherein said impedance element constituted that an inductor is connected in series to a feed back resistor of an inverse amplifier of said second stage.
 3. The biquad notch filter claimed in claim 1 , wherein said impedance element constituted that a capacitor is connected in parallel to a feed back resistor of an inverse amplifier of said second stage.
 4. The biquad notch filter claimed in claim 1 , wherein said impedance element constituted that a capacitor is connected to a feed back resistor in a parallel and an inductor is connected to them in series. 